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arXiv · 2309.13616

Spectral estimates of the Dirichlet-Laplace operator in conformal regular domains

Ivan Kolesnikov·Valerii Pchelintsev
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Abstract

In this paper we consider conformal spectral estimates of the Dirichlet-Laplace operator in conformal regular domains $\Omega \subset \mathbb R^2$. This study is based on the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincar\'e-Sobolev inequalities. On this base we obtain lower estimates of the first eigenvalue of the Dirichlet-Laplace operator in a class of conformal regular domains. As a consequence we obtain conformal estimates of the ground state energy of quantum billiards.

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Ivan Kolesnikov, Valerii Pchelintsev. 2023-09-24. Spectral estimates of the Dirichlet-Laplace operator in conformal regular domains. https://arxiv.org/abs/2309.13616

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